#72312292

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#10991
commutator	C₂²×D₄	C₂×C₄²
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#123590
prime radical	C₂×#3449	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂² ▷ C₂ ▷ C₁

4933 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,11,8,12,6,13)(3,15,17,34,20,9,18,33)(4,10,22,37,23,39,25,40)(7,28,26,42,31,14,27,41)(16,45,46,57,47,32,48,56)(19,43,49,60,29,35,50,59)(21,36,51,61,52,58,55,64)(24,44,53,63,30,38,54,62);(1,3)(2,9)(4,21)(5,17)(6,18)(7,29,31,19)(8,20)(10,36)(11,33)(12,15)(13,34)(14,43,28,35)(16,30,47,24)(22,51)(23,52)(25,55)(26,50,27,49)(32,38,45,44)(37,61)(39,58)(40,64)(41,60,42,59)(46,54,48,53)(56,62,57,63);(1,4,5,25,8,23,6,22)(2,10,13,37,12,39,11,40)(3,16,18,46,20,47,17,48)(7,30,26,53,31,24,27,54)(9,32,33,57,15,45,34,56)(14,44,42,63,28,38,41,62)(19,21,50,51,29,52,49,55)(35,58,59,61,43,36,60,64)

Polycyclic presentation:

2^10|a^2=d,b^2=h,c^2=dei,d^2=j,f^2=j,g^2=j,b^a=bdfj,c^a=ce,c^b=cdefgj,d^c=dj,e^a=ej,e^c=ej,f^c=fi,g^a=gij,g^b=gi,g^c=gij,g^f=gj,h^c=hi,h^g=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312305, #72312888, #72313192
maximal normal subgroup of	#72312305, #72312888, #72313192

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.4*F.5*F.9);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.6*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5*F.6*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);